Rui Loja Fernandes

On the master symmetries and bi-Hamiltonian structure of the Toda lattice

Starting from a conformal symmetry, higher-order Poisson tensors, deformation relations and master symmetries for the Toda lattice are obtained. A hierarchy of time-dependent symmetries is also constructed. Using reduction, deformation relations previously known to hold up to a certain equivalence relation are shown to be exact.

FROM THE TODA LATTICE TO THE VOLTERRA LATTICE AND BACK

We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices.

Completely integrable bi-Hamiltonian systems

We study the geometry of completely integrable bi-Hamiltonian systems and, in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the affine structure determined by the action variables. This generalizes a result of Brouzet for dimension four.

Riemannian metrics on Lie groupoids

We introduce a notion of metric on a Lie groupoid, compatible with multipli- cation, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allows us to establish a linearization theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein-Zung linearization theorem for proper Lie groupoids. This new notion of metric has a simplicial nature which will be explored in future papers of this series.

Integrability of the periodic KM system

Abstract We study the integrability of the periodic Kac-van Moerbeke system. We give a bi-Hamiltonian formulation and a Lax pair containing a spectral parameter. Using Griffiths aproach we linearize the system on the Jacobian of the associated spectral curve.

Poisson Manifolds of Compact Types (PMCT 1)

Abstract This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.

Invariants of Lie algebroids

Abstract Several new invariants of Lie algebroids have been discovered recently. We give an overview of these invariants and establish several relationships between them.

The classifying Lie algebroid of a geometric structure I: Classes of coframes

We present a systematic study of symmetries, invariants and moduli spaces of classes of coframes. We introduce a classifying Lie algebroid to give a complete description of the solution to Cartans realization problem that applies to both the local and the global versions of this problem.

Integration of coupling Dirac structures

Coupling Dirac structures are Dirac structures defined on the total space of a fibration, generalizing hamiltonian fibrations from symplectic geometry, where one replaces the symplectic structure on the fibers by a Poisson structure. We study the associated Poisson gauge theory, in order to describe the presymplectic groupoid integrating coupling Dirac structures. We find the obstructions to integrability and we give explicit geometric descriptions of the integration.

Riemannian metrics on differentiable stacks

We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.